> > Are the following statements "mathematics" by your definition? Why or why not?
> > 1) White wins in chess if you remove Black's Queen from the initial position.
>> YES IT'S ABOUT SOMETHING NONPHYSICAL (ABSTRACT) AND HAS A
> CONVINCING, ACCEPTED PROOF.
Indeed! But the proof is NOT a mathematical one!
An aside to state those of my credentials relevant to the following:
I currently compete in correspondence chess at a high level, and made
a plus score in the most recently completed United States
Championship (Invitational). My peak rating is 2405 (Senior Master)
and I have also had a Master's rating in ordinary tournament chess.
>From 1987 to 1991 I was a regular chess writer for Chess Horizons
Magazine on the theory of Chess Openings and won four awards from the
Chess Journalists of America for this work (including two for "Best
Regular Magazine Column" and one for "Best Analysis").
Every chessplayer with significant practical experience knows that
the advantage of a Queen in the initial position is enough to win.
This is truly a known fact, in the philosophical sense of justified
true belief. It is also clearly a mathematically formalizable
statement. But this fact is not (yet) mathematical knowledge because
we have nothing even remotely approaching a mathematical proof.
It seems, from his response, that Hersh mistakenly thinks we have
such a proof. But this isn't necessarily so. Recall that his
definition of mathematics is simply that area of abstract human
thought which is uniquely objective, characterized by the full
reproducibility of and social consensus about its results. (Other
sciences don't qualify because they are not abstract, other
humanities don't qualify because there is no reproducibility and
consensus.) Chess is abstract. There is a full consensus among
everyone competent in the subject (let us say, everyone with a
rating equivalent to 1600 Elo points or higher; there are hundreds of
thousands if not millions of such players worldwide) about this
result. The result is reproducible in a very strong sense (if you
take two players who are roughly equal in strength and make them play
a "Queen-odds" game the player with the Queen will always win).
Even if the successor to Deep Blue could somehow prove by exhaustive
search that White wins when you remove Black's Queen (which I don't
think will ever happen), Hersh's definition also applies to much
more complex results, such as "The initial position is not a forced
win for the second player". No machine could ever establish this,
and I doubt it will ever be shown in a mathematically rigorous sense
(I think it is more likely that it will be rigorously shown that any
proof of that statement would be astronomical in size). But you
again have consensus and reproducibility (ask any good player to
choose a color in a money game against an unidentified opponent and
he will choose White; I specify "unidentified" because against a
specific opponent he might have a surprise Black opening prepared).
Hersh's definition fails because it is purely external; there is
something about the nature of mathematical proofs that distinguishes
them from Hershian ones. To fully "mathematize" proofs of statements
like 1) above for games like Chess or Go is a fascinating research
program, but so far progress has only been made in endgames with so
few pieces that an exhaustive search of all positions is possible.
Unless and until this hugely ambitious program is accomplished (I
know it is on Harvey's list of things to do and he'll undoubtedly get
to it within a few centuries), I maintain that Hersh's definition of
what mathematics "really" is invalid and challenge everyone on the
FOM list who agrees with this to come up with a better definition
mathematics which properly distinguishes mathematical proof from
the kind of proof we have of statement 1).
By the way, Reuben, I intend to argue in a later posting for the
expansion of the notion of proof to include what computer scientists
call "interactive proofs", which will in fact make my chess example
somewhat closer to what is to be regarded as mathematically provable
and partially vindicate your definition (extensionally in the sense
that I can't think of any better counterexamples, not intensionally
because your definition is external and has nothing to say about the
internal nature of proofs).
-- Joe Shipman